Consider the system of linear differential equations (d 𝐱)/(d t)=[ -2 -1 2 1 ] 𝐱 Th

step 1 Okay, so in this problem, we're given this system of differential equations. So for part A, it's

Consider the system of linear differential equations (d...

Consider the system of linear differential equations $$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-2} & {-1} \\ {2} & {1}\end{array}\right] \mathbf{x}$$ The system is nongeneric, that is, the determinant of the matrix of coefficients is zero. (a) There are an infinite number of equilibria, all lying on a line in the phase plane. What is the equation of this line? (b) Construct the phase plane for this system.

Consider the system of linear differential equations

$$\frac{d \mathbf{x}}{d t}=\left[ \begin{array}{rr}{-2} & {-1} \\ {2} & {1}\end{array}\right] \mathbf{x}$$

The system is nongeneric, that is, the determinant of the matrix of coefficients is zero.
(a) There are an infinite number of equilibria, all lying on a line in the phase plane. What is the equation of this line?
(b) Construct the phase plane for this system.

Key Concepts

Systems of Linear Differential Equations

A system of linear differential equations with constant coefficients represents a set of first?order equations where the rate of change of each variable is given by a linear combination of the variables. These systems are fundamental in modeling many natural phenomena and allow analysis of dynamics through matrix methods.

Equilibrium Points in Dynamical Systems

Equilibrium points are the states at which the system does not change over time; they are found by setting the derivative (or system function) equal to zero. In linear systems, determining equilibria involves solving a system of homogeneous linear equations, which can yield an isolated point or a continuum of points.

Singular Matrices and Nongeneric Systems

A singular matrix, one with a zero determinant, lacks full rank and therefore does not map the entire space injectively. In the context of differential equations, a singular coefficient matrix leads to nongeneric behavior such as the presence of infinitely many equilibria, because the system of equations becomes under-determined.

Nullspace of a Matrix and Lines of Equilibria

The nullspace (or kernel) of a matrix consists of all vectors that are mapped to the zero vector. When a system’s coefficient matrix is singular, its nullspace is non-trivial, forming a subspace (often a line or a plane) of equilibrium solutions. This explains the infinite number of equilibria lying on a specific geometric structure in the phase plane.

Phase Plane Analysis

Phase plane analysis is a graphical tool used to study the behavior of two-dimensional dynamical systems. By plotting trajectories, identifying equilibrium points, and determining their stability, this method helps visualize how solutions evolve over time, particularly illustrating critical features like invariant lines of equilibria.

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